Chapter VI: Integration of Jordan structures

Abstract

In Chapter III we established the equivalence of Jordan algebraic structures with certain geometric objects (symmetric spaces with additional structure). However, the method used there does not give much insight into the geometry of the symmetric spaces in question. If we want to understand the geometry, we have to pass from tensors to the spaces itself. The fundamental tool to be used in this process is the integrability of invariant almost complex structures and polarizations on symmetric spaces. The fact that such structures are integrable is well-known and easy to prove (we don’t need to invoke the theorem of Newlander and Nirenberg; cf. Appendix VLA), but it has important consequences.